I think it has been made clear already, but here's my two cents:

An acoustic guitar is one that needs not be amplified in order to produce its native tone, as opposed to, say, a Fender Stratocaster, which thing has its own tone only to the extent that it is amplified.

An acoustic guitar with a pickup in it is not, IMO, the same thing as a strictly acoustic guitar, even of the same model. They often have completely different tonalities.

Acoustic music is music that is created by strictly acoustic instruments. Amplification of an acoustic instrument (like a mic in front of an acoustic guitar) is just that, amplification of ACOUSTIC music. The sound that comes through a PA system of a plugged-in guitar is not acoustic music, it's electric music or at least acoustic-electric music.

Take this for example:

Bill Monroe and his Blue Grass Boys: ACOUSTIC band, all gathered around a or several microphones which were used to amplify strictly acoustic music.

The later incarnations of Flatt & Scruggs and the Foggy Mountain Boys: at most, an electric-acoustic band, as they almost all adopted plugs for their instruments as the technology became available. They were also clear in many interviews about how they liked what the pickups and so on did to change the tonality of their instruments.

Anyway, I'm more of a purist about these things.

Sing along,

Steve